Secondary Math 3 H - Section 2.5 - Zeros of Polynomial Functions
Starter Question- Factor the polynomial and use the leading coefficient and the zeros and their multiplicity to sketch the graph of
In section 2.2 we learned that finding the factors and the zeros of a polynomial can make graphing a lot easier. We also learned that a polynomial of degree can have at most _________ zeros. If we allow for complex zeros, we can we can say that a polynomial of degree has exactly __________ zeros. There are theorems in section 2.5 that will make finding these zeros a little easier. Most of these theorems don’t actually tell us what the zeros are (that’s what the remainder and factor theorems and synthetic division from section 2.3 are for), but they will help narrow our search.
Fundamental Theorem of Algebra-
If is a polynomial of degree , where , then has at least one zero in the complex number system.
In other words- We are guaranteed at least one complex _________.
Real Zeros of an Odd Degree Polynomial-
A polynomial function with real coefficients of odd degree has at least one real zero.
In other words- If the polynomial is of _____ degree, we are guaranteed at least one real _______.
Example 1- a. Explain or draw a picture to why this is true?
b. Do we have the same guarantee if the polynomial is even? Why or why not?
Number of Real Zeros-
A polynomial function cannot have more real zeros that its degree.
In other words- This tells us how many ________ zeros there can be and when we can stop looking.
Example 2- How many real zeros can the following functions have?
a. b. c.
Number of Complex Zeros-
A polynomial function of degree , has precisely complex zeros
In other words- This tells us how exactly many _________ zeros we’re looking for and when we can stop.
Linear Factorization Theorem-
If is a polynomial of degree , where , then has precisely linear factors written as: where , , …, are complex numbers.
In other words- We are guaranteed that every polynomial can be ____________ using complex numbers.
Example 4- Factor the polynomial completely and find the zeros:
This was a relatively simple example, now we’ll look at more complicated factoring.
Polynomial Factorization-
Every polynomial function with real coefficients can be written as the product of linear factors and irreducible quadratic factors.
In other words- We can factor every polynomial using linear factors like , which are linear, or, irreducible quadratics, like that don’t have any real solutions. Finding the zeros from linear factors is easy. Find zeros from irreducible quadratics can be done using the __________ formula.
Example 5- Factor the polynomial completely and find the zeros:
The last problem had some pretty ugly answers. If we have a choice, it would be good to find the nice zeros first.
Rational Zeros Theorem- a.k.a.- the “ ___ and ___ stuff” If the polynomial
has integer coefficients, then every rational zero of has the form , where is a factor of and is a factor of .
In other words- and will give us a list of “nice” numbers to check first. Please note that and will give a list possible zeros, we still need to see which ones are zeros.
Example 6- List the possible rational zeros of
Conjugate Pairs Theorem-
If is a polynomial with real coefficients, and is a zero of , then is also a zero of .
In other words- Imaginary zeros always come in ___________ pairs.
Example 7- Can you name any other zeros of if two of the zeros are and ?
What is the lowest degree polynomial that can have these zeros?
Section 2.5 also mentions two other theorems that we WILL NOT use: Descarte’s Rule of ___________ and The Upper and Lower _________ Theorem. They are useful, but in the interest of time and in aligning with SLCC, we will not use them.
As you do the problems, pay attention to the instructions. If it says find all the real zeros or factor over the real numbers, you can only use real numbers, no ’s. If it says find ALL zeros or factor over the complex numbers, then you can use numbers including the imaginary numbers.
Please consider the problems in the notes part of your homework for this section.
Example 8- Find the real zeros of the following:
a. b.
Example 9- Find all the zeros of the following:
a. b.
Look at the quadratics in the last two examples. Do you see a difference between them?
What if we were asked to completely factor the polynomials above?
Example 10- Use the Rational Zeros Test to list all possible rational zeros of . Verify the zeros shown on the graph are on the list of possibilities.
Example 12- Solve the equation over the real number system:
Example 13- Find all the zeros of the function
Example 14- Use the information to find the remaining zeros of the polynomial.
a. Degree 4, with zeros: , b. Degree 6, with zeros: , , ,
Example 16- Find the complex zeros of the polynomial function then write the function in factored form. (Think hard on this one. Put all the ideas together.)
Example 18- Find the complex zeros of each polynomial function then write the function in factored form.